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In the usual way of constructing the cumulative hierarchy under the iterative conception of sets, sets are constructed in levels. George Boolos, in his "The Iterative Conception of Set", devises a "stage theory" where each level/stage is akin to a possible world in modal logic and there is an "accessibility relation" that relates each stage to its predecessor stages. One popular way of understanding accessibility in this context is as a relation of dependence so that each stage depends on its predecessor stages.

Obviously this view of things isn't forced on us, but grant for the sake of the question this idea that the levels of the hierarchy "depend" on preceding levels. My question is whether there are other mathematical theories that seem to exhibit this same sort of "dependence" phenomena. For an intuitionist, just about any theory has this flavor since an object can be said to "depend" on any objects used in its construction. But are there examples from non-explicitly intuitionistic mathematical theories that seem to have this dependence/modal feature?

Dennis
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  • One sense of "iterative conception" might be any recursive definition. For example, polynomials can be defined by recursion (or inductively) starting from constants as a basis step and proceeding to build up polynomials from simpler ones to more complex ones by adding and multiplying. Such recursive definitions are ubiquitous in mathematics. Is there some further purpose to your Question that might point Answers in the direction of "intuitionistic mathematical theories"? – hardmath Dec 20 '19 at 01:15
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    Surreal numbers are quite explicit about this sort of relation (i.e., birthdays). – Rushabh Mehta Dec 20 '19 at 01:20
  • @hardmath I’m interested in the natural language semantics of mathematics, e.g., formal semantics for the sort of text you might encounter in a proof that isn’t totally formalized. I’m especially interested in anaphora in mathematics as well as modal claims about mathematical objects. For example, if you are talking about the complex numbers you have two objects that could equally well serve as the referent of “i_”. It doesn’t matter which you assign to _i, but once one has been chosen then the other candidate must be the referent of $-i$ — there’s a sort of dependence between the terms. – Dennis Dec 20 '19 at 01:31
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    Direct limits in category theory are a direct extension of iterative constructions. In general, the idea of filtration as "cumulative hierarchy" is ubiquitous, from homology complexes in algebra to martingales in probability. There is Borel hierarchy, arithmetical hierarchy, Grzegorczyk hierarchy, even hierarchy of spaces, etc. – Conifold Dec 20 '19 at 01:54

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